What number is 110% of 150?

Set up a proportion

`x` = the 110% of 150 the part
150 = the total parts y
110% = the percent part
100% = the total percent.

Using the principle of ratio I choose to set 100 as the equivalent of the original quantity of 150 and 110 as the equivalent of the new unknown quantity.

Let `x` be the new unknown quantity giving the ratio

`("New quantity")/("original quantity") =110/100=x/150`

Multiply both sided by 150

`( 110 xx 150)/ 100 = ( x xx 150)/150`

This gives

`16500/100 = x`

`165 = x`

`color(blue)("The underlying principle of percentage")`

`color(red)("It just a fraction.")` However, it is a special fraction because the bottom number (denominator) is fixed at 100.

The symbol % behaves like a unit of measurement but one that is worth `xx1/100` including the multiply sign

So `110%` is the same as `110xx1/100 = 110/100`

The question states `110% color(red)(" of ") 150`

Translating the word 'of' into mathematics it becomes multiply

`color(green)(110color(white)("dd")color(blue)(%)color(white)("dd.")color(red)("of")color(white)("dd")150)`

`color(green)(110color(white)("dd")color(blue)(%)color(white)("dd.")color(red)(xx)color(white)("d")150)`

`color(green)(110color(blue)(obrace(xx1/100))color(white)("d")color(red)(xx)color(white)("d")150)`

Giving:

`110/100xx150 larr" Starting point"`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("A trick to calculating this in the head")`

`110/100 -> 100/100+10/100 = 1 +1/10`

`color(white)("d")`

`color(white)("d")1color(white)(".")xx150=150`
`1/10xx150= ul(color(white)(1)15larr" Add")`
`color(white)("dddddddddd")165`